${\rm PD}\sb 4$-complexes with free fundamental group

We investigate the geometry of closed, orientable, hyperbolic 3-manifolds whose fundamental groups are [Formula: see text]-free for a given integer [Formula: see text]. We show that any such manifold [Formula: see text] contains a point [Formula: see text] with the following property: If [Formula: see text] is the set of maximal cyclic subgroups of [Formula: see text] that contain non-trivial elements represented by loops of [Formula: see text], then for every subset [Formula: see text], we have rank [Formula: see text]. This generalizes to all [Formula: see text] results proved in [J. W. Anderson, R. D. Canary, M. Culler and P. B. Shalen, Free Kleinian groups and volumes of hyperbolic 3-manifolds, J. Differential Geom. 43 (1996) 738–782; M. Culler and P. B. Shalen, 4-free groups and hyperbolic geometry, J. Topol. 5 (2012) 81–136], which have been used to relate the volume of a hyperbolic manifold to its topological properties, and it strictly improves on the result obtained in [R. K. Guzman, Hyperbolic 3-manifolds with [Formula: see text]-free fundamental group, Topology Appl. 173 (2014) 142–156] for [Formula: see text]. The proof avoids the use of results about ranks of joins and intersections in free groups that were used in [M. Culler and P. B. Shalen, 4-free groups and hyperbolic geometry, J. Topol. 5 (2012) 81–136; R. K. Guzman, Hyperbolic 3-manifolds with [Formula: see text]-free fundamental group, Topology Appl. 173 (2014) 142–156].

We classify compact oriented 5-manifolds with free fundamental group and π 2 a torsion free abelian group in terms of the second homotopy group considered as π 1 -module, the cup product on the second cohomology of the universal covering, and the second Stiefel-Whitney class of the universal covering.We apply this to the classification of simple boundary links of 3-spheres in S 5 .Using this we give a complete algebraic picture of closed 5manifolds with free fundamental group and trivial second homology group.

Schreier proved in [8] tha t a finitely generated normal subgroup U # { 1 } of a free group F is of finite index. This result was extended by Karrass and Solitar in [3], to the case when U is not necessarily normal, but contains a non-trivial normal subgroup of F. In Topology, the free groups occur as fundamental groups of surfaces with boundary, and we here extend the result still further (Theorem 6.1) to the case when F is the (non-abelian) fundamental group of any connected surface, with or without boundary, except for a Klein bottle. We use topological methods, and also the elements of Morse theory, although the latter could be ehminated. A sketch of this theory is included, however, par t ly for its intuitive appeal, and part ly because the Morse theory picks out stable generators of the fundamental group, and therefore is helpful as a tool. Indeed, the author was able to use it quickly to prove Schreier's Theorem, and Theorem 3.3 below (that an open surface has free fundamental group), before knowing tha t proofs already existed in the literature. Our exposition is always from the point of view tha t it is the surface, ra ther than the group, which is the ult imate object o f study; and we have perhaps laboured points tha t might irk the pure group-theorist.

We calculate the group of homotopy classes of homotopy self-equivalences of 4-manifolds with free fundamental group and obtain a classification of such 4-manifolds up to s-cobordism.

Let $\Lambda$ be a ring endowed with an involution $a \mapsto \tilde{a}$. We say that two units $a$ and $b$ of $\Lambda$ fixed under the involution are congruent if there exists an element $u \in \Lambda^{\times}$ such that $a = ub\tilde{u}$. We denote by $\mathcal{H}(\Lambda)$ the set of congruence classes. In this paper we consider the case where $\Lambda$ is an order with involution in a semisimple algebra $A$ over a local field and study the question of whether the natural map $\mathcal{H}(\Lambda) \rightarrow \mathcal{H}(\Lambda)$ induced by inclusion is injective. We give sufficient conditions on the order $\Lambda$ for this map to be injective and give applications to hermitian forms over group rings.

On the regular genus of 5-manifolds with free fundamental group

Manifolds with free abelian fundamental groups and their applications (Pontrjagin classes, smoothnesses, multidimensional knots)

The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G) using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this thesis we introduce natural subspaces B(q,G)_p of B(q,G) defined for a fixed prime p. We show that B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)_p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial p-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial p-groups of rank at least 4 the space B(2,G) does not have the homotopy type of a K(π,1) space. Furthermore, we give a group theoretic condition, applicable to symmetric groups and general linear groups, which implies the space B(2,G) not having the homotopy type of a K(π,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.

Using the wonderful compactification of a semisimple adjoint affine algebraic\ngroup G defined over an algebraically closed field k of arbitrary\ncharacteristic, we construct a natural compactification Y of the G-character\nvariety of any finitely generated group F. When F is a free group, we show that\nthis compactification is always simply connected with respect to the \\'etale\nfundamental group, and when k=C it is also topologically simply connected. For\nother groups F, we describe conditions for the compactification of the moduli\nspace to be simply connected and give examples when these conditions are\nsatisfied, including closed surface groups and free abelian groups when\nG=PGL(n,C). Additionally, when F is a free group we identify the boundary\ndivisors of Y in terms of previously studied moduli spaces, and we construct a\nfamily of Poisson structures on Y and its boundary divisors arising from\nBelavin-Drinfeld splittings of the double of the Lie algebra of G. In the\nappendix, authored by Sam Evens and Arlo Caine, we explain how to put a Poisson\nstructure on a quotient of a Poisson algebraic variety by the action of a\nreductive Poisson algebraic group.\n

The Whitehead asphericity conjecture claims that if ⟨ A ‖ R ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {R} \, \rangle is an aspherical group presentation, then for every S ⊂ R \mathcal {S} \subset \mathcal {R} the subpresentation ⟨ A ‖ S ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {S} \, \rangle is also aspherical. This conjecture is generalized for presentations of groups with periodic elements by introduction of almost aspherical presentations. It is proven that the generalized Whitehead asphericity conjecture (which claims that every subpresentation of an almost aspherical presentation is also almost aspherical) is equivalent to the original Whitehead conjecture and holds for standard presentations of free Burnside groups of large odd exponent, Tarski monsters and some others. Next, it is proven that if the Whitehead conjecture is false, then there is an aspherical presentation E = ⟨ A ‖ R ∪ z ⟩ E = \langle \, \mathcal {A} \, \| \, \mathcal {R} \cup z \, \rangle of the trivial group E E , where the alphabet A \mathcal {A} is finite or countably infinite and z ∈ A z \in \mathcal {A} , such that its subpresentation ⟨ A ‖ R ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {R} \, \rangle is not aspherical. It is also proven that if the Whitehead conjecture fails for finite presentations (i.e., with finite A \mathcal {A} and R \mathcal {R} ), then there is a finite aspherical presentation ⟨ A ‖ R ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {R} \, \rangle , R = { R 1 , R 2 , … , R n } \mathcal {R} = \{ R_{1}, R_{2}, \dots , R_{n} \} , such that for every S ⊆ R \mathcal {S} \subseteq \mathcal {R} the subpresentation ⟨ A ‖ S ⟩ \langle \, \mathcal {A} \, \| \, \mathcal {S} \, \rangle is aspherical and the subpresentation ⟨ A ‖ R 1 R 2 , R 3 , … , R n ⟩ \langle \, \mathcal {A} \, \| \, R_{1}R_{2}, R_{3}, \dots , R_{n}\, \rangle of aspherical ⟨ A ‖ R 1 R 2 , R 2 , R 3 , … , R n ⟩ \langle \, \mathcal {A} \, \| \, R_{1}R_{2}, R_{2}, R_{3}, \dots , R_{n}\, \rangle is not aspherical. Now suppose a group presentation H = ⟨ A ‖ R ⟩ H = \langle \, \mathcal {A} \, \| \, \mathcal {R} \, \rangle is aspherical, x ∉ A x \not \in \mathcal {A} , W ( A ∪ x ) W(\mathcal {A} \cup x) is a word in the alphabet ( A ∪ x ) ± 1 (\mathcal {A} \cup x)^{\pm 1} with nonzero sum of exponents on x x , and the group H H naturally embeds in G = ⟨ A ∪ x ‖ R ∪ W ( A ∪ x ) ⟩ G = \langle \, \mathcal {A} \cup x \, \| \, \mathcal {R} \cup W(\mathcal {A} \cup x) \, \rangle . It is conjectured that the presentation G = ⟨ A ∪ x ‖ R ∪ W ( A ∪ x ) ⟩ G = \langle \, \mathcal {A} \cup x \, \| \, \mathcal {R} \cup W(\mathcal {A} \cup x) \, \rangle is aspherical if and only if G G is torsion free. It is proven that if this conjecture is false and G = ⟨ A ∪ x ‖ R ∪ W ( A ∪ x ) ⟩ G = \langle \, \mathcal {A} \cup x \, \| \, \mathcal {R} \cup W(\mathcal {A} \cup x) \, \rangle is a counterexample, then the integral group ring Z ( G ) \mathbb {Z}(G) of the torsion free group G G will contain zero divisors. Some special cases where this conjecture holds are also indicated.

We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various sub- groups of the mapping class group—the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration—and additionally to the outer automorphism group of a free group and to certain linear groups. There are many analogies and interconnections between the theories of right- angled Artin groups on one hand and mapping class groups on the other hand. For instance, by the work of Crisp and Wiest (13), the work of Koberda (27), and the work of the first two authors with Mangahas (12), there is an abundance of injective homomorphisms from right-angled Artin groups to mapping class groups. Also, the last two authors proved (30) that any two elements of the pure braid group either generate a free group or a free abelian group—a property shared by all right-angled Artin groups (4, Theorem 1.2). We are thus led to ask to what extent mapping class groups are the same as right-angled Artin groups. It is straightforward to see that most mapping class groups are not isomorphic to right-angled Artin groups, for instance because right-angled Artin groups are torsion free. On the other hand, mapping class groups have finite-index sub- groups that are torsion free, and so this leaves open the possibility that mapping class groups are abstractly commensurable to right-angled Artin groups, that is, that they have isomorphic finite-index subgroups. We prove that, aside from a small number of exceptions, this is not the case. We also extend this result to several classes of groups related to mapping class groups. We start by recalling some definitions. To a finite graph , we can associate a right-angled Artin grou p: this is the group with one generator for each vertex of , and one defin ing relator for each edge, namely, the commutator of the two generators corresponding to the endpoints. Let Sg,n denote a closed, connected, orientable surface of genus g with n marked points. The mapping class group Mod(Sg,n) is the group of homotopy classes of orientation-preserving homeomorphisms of Sg,n preserving the set of marked points. As discussed in Koberda's paper (27, Theorem 1.5), no finite-index subgroup of Mod(Sg,n) injects into a right-angled Artin group if g ≥ 2 and (g,n) 6 (2,0); see

Class group of the group ring of a cyclic group of order p2 over the ring of integers in a quadratic number field

Nielsen fixed point theory for split n-valued maps on the Klein bottle

Some primitive group rings

A BV-algebra structure on Hochschild cohomology of the integral group ring of finitely generated Abelian groups